I am trying to recover fully a proof of a result which is attributed to M. Hopkins in the litterature, namely the following:
Let $h:S^1\to BGL_1(\mathbb S_p)$ detect the element $1-p$, then the Thom spectrum of the induced map $f:\Omega^2S^3\to BGL_1(\mathbb S_p)$ is $H\mathbb F_p$
The proof I'm trying to focus on is that of Blumberg in THH of Thom spectra that are $E_\infty$-ring spectra. After showing that $Mf$ has a Thom class mod $p$, the author says that the argument is basically the same as the argument for $p=2$ where the sphere spectrum need not be localized.
The argument to which he is referring is that of Priddy in $K(Z/2,0)$ as a Thom spectrum : $H_*(\Omega^2 S^3;\mathbb F_2)\to \mathcal A_2^*$ is non trivial hence an isomorphism in degree $1$. Also, we know that $H_*(\Omega^2 S^3;\mathbb F_2)$ is generated by its degree one generator and the Dyer Lashof operation $Q_1$, as an algebra, and that the Thom isomorphisms preserve Dyer Lashof operations; and we know the action of $Q_1$ on the homology of the universal Thom spectrum $MO$ : $Q_1^{\circ n}p_1=p_n$ where $p_i$ is the primitive element in that dimension. So, the morphism of interest hits all the $p_i$s. Then by computing appropriate Kronecker brackets it is easy to show that the post composition with $H_*(MO;\mathbb F_2)\to \mathcal A_2^*$ is surjective hence an isomorphism.
I don't see how a proof for odd primes can be the same as that, because in our case the map does not factor (a priori) to an equivalent of $MO$ in the localized setting. Would it be that the homology of $MGL_1(\mathbb S^p)$ is well known as well as the action of Dyer Lashof operations on it ? So here is my main question : what does Blumberg mean by "an analoguous argument" in his proof ?
Otherwise, I can guess that there is a simpler proof in the case $p=2$ : as $f$ is an $E_2$ map, $Mf$ is an $E_2$-ring spectrum and it would not be surprising (although I don't know how to show it) if the map $Mf\to H\mathbb F_2$ representing the Thom class was an $E_2$-map. I know a proof of that in the $E_1$ case, where it is enough to work in the homotopy category. Would this be true for any connective $E_n$-ring spectrum ? Anyway, if it is true in our $E_2$ case, then it is just a matter of knowing the Dyer Lashof action on $H\mathbb F_2$, and the proof is then automatic. Also, the proof would be the same for odd primes.
Thank you for your help.
[EDIT] Indeed if $X$ is a connective $E_n$-ring spectrum, then the canonical map $E\to H(\pi_0(E))$ is an $E_n$-map because the two maps $$E_n(k)_+\wedge_{\mathfrak S_n} E^{(k)}\to\cdots\to H(\pi_0(E))$$ of interest both induce the multiplication map $\pi_0(E)^r\to\pi_0(E)$. So if I'm not mistaken the proof can be finished just by looking at the Dyer Lashof operations on $H\mathbb F_p$ (by using this fact for $E:=Mf\wedge H\mathbb F_p)$.